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Subject: Re: More complex numbers than =?UTF-8?Q?reals=3F?=
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Date: Wed, 10 Jul 24 16:14:20 +0000
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From: WM <wolfgang.mueckenheim@tha.de>
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Le 10/07/2024 à 01:45, Ben Bacarisse a écrit :
> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
> 
>> On 7/9/2024 10:30 AM, Ben Bacarisse wrote:
>>> WM <wolfgang.mueckenheim@tha.de> writes:
>>> 
>>>> Le 09/07/2024 à 14:37, Ben Bacarisse a écrit :
>>>>
>>>>> A mathematician, to whom this is a whole new topic, would start by
>>>>> asking you what you mean by "more".  Without that, they could not
>>>>> possibly answer you.
>>>>
>>>> Good mathematicians could.
>>>>
>>>>>   So, what do you mean by "more" when applied to
>>>>> sets like C and R?
>>>>
>>>> Proper subsets have less elements than their supersets.
>>>
>>> Let's see if Chris is using that definition.  I think he's cleverer than
>>> you so he will probably want to be able to say that {1,2,3} has "more"
>>> elements than {4,5}.
>>
>> I was just thinking that there seems to be "more" reals than natural
>> numbers. Every natural number is a real, but not all reals are natural
>> numbers.
> 
> You are repeating yourself.  What do you mean by "more"?  Can you think
> if a general rule -- a test maybe -- that could be applied to any two
> set to find one which has more elements?

No rule is better than a foolish rule, if it yields nonsense like Cantor's 
"bijections".
> 
>> So, wrt the complex. Well... Every complex number has a x, or real
>> component. However, not every real has a y, or imaginary component...
>>
>> Fair enough? Or still crap? ;^o
> 
> So you are using WM's definition based on subsets?  That's a shame.

It is a reliable rule.

> One consequence is that you can't say if the set of even numbers has
> more or fewer elements than {1,3,5} because {1,3,5} is not a subset of
> the even numbers, and the set of even numbers is not a subset of
> {1,3,5}.  They just can't be compared using your (and WM's) notion of
> "more".

There are further rules. Every finite set has less elements than an 
infinite set. The set of rational numbers has 2|N|^2 + 1 elements. The set 
of real numbers is much larger than the set of rational numbers (but not 
because of Cantor's nonsense).

Regards, WM